Tuning the non-linear interactions of hybrid interlayer excitons in bilayer MoS2 via electric fields

TL;DR

The paper addresses how to control nonlinear exciton–exciton interactions in hybrid interlayer excitons of bilayer MoS$_2$ using an external out-of-plane electric field. A microscopic framework combining first-principles band structures with Coulomb matrix elements computes density-induced energy renormalizations through a self-consistent self-energy $\tilde{E}_{\nu,\mathbf{Q}} = E_{\nu,\mathbf{Q}} + \mathrm{Re}\,\Sigma(\nu,\mathbf{Q},\tilde{E}_{\nu,\mathbf{Q}}/\hbar)$ where $\Sigma = \Sigma^{\mathrm{H}} + \Sigma^{\mathrm{F}} + \Sigma^{\mathrm{PB}} + \Sigma^{\mathrm{MW}}$. Experimentally, field-polarized hIEs exhibit a substantially larger density-dependent blueshift than at zero field, consistent with enhanced dipolar repulsion dominating over attractive many-body corrections. The results demonstrate tunable strong nonlinearities in excitons without relying solely on density, providing a pathway toward electrically controlled excitonic polaritons and related devices. The work thus establishes bilayer MoS$_2$ as a robust platform for engineering dipolar exciton interactions with direct implications for future optoelectronic technologies.

Abstract

Hybrid interlayer excitons in bilayer MoS2 are a promising platform for nonlinear optics due to their intrinsic dipolar character, which combines in-plane and out-ofplane dipole moments. In this work, we directly probe the nonlinear exciton-exciton interactions of hybrid interlayer excitons. By applying an external out-of-plane electric field, we polarize the excitons to enhance their mutual dipolar interactions, thereby deliberately favoring these repulsive contributions over competing attractive manybody corrections. We furthermore establish a fully microscopic theoretical description of these effects to explain the core experimental results. The tuning results in a significantly larger blueshift compared to the zero-field case and perspectively opens an avenue to even switch between repulsive and attractive interaction potentials. Our findings establish that strong nonlinearities can be tuned via an external electric field, providing a new degree of control over exciton interactions beyond density tuning alone.

Figures (5)

• Figure 1: (a) Real-space structure of the emergent exctions in bilayer MoS$_2$, composed of intra- as well as hybrid interlayer excitons. (b) Band structure illustration of the A interlayer exciton and the hIE around the K point. The tunneling of the hole between the two layers is indicated by the green arrow. In (c) we vary a voltage across the two graphene layers and visualize the splitting of the hIX by measuring the reflection contrast.
• Figure 2: (a) Reflectivity measurements of bilayer MoS$_2$ at zero gate voltage. Increasing the power of an additional green laser from 0 $\mu$W to 240 $\mu$W results in a net blue shift of the two exciton peaks. The peak around 1.94 eV corresponds to the intra-layer A exciton, while the peak at 2.01 eV is identified as the hIE. (b) Extracted energy shifts of the A exciton and the hIE exciton versus the density of the respective exciton species.
• Figure 3: (a) Reflectivity contrast measurements at 1 V gate voltage, corresponding to a local electric field at the bilayer of roughly 0.1 V/nm. The A exciton peak is at 1.94 eV, while the hIE splits into a lower-energy state at 1.99 eV and a higher-energy state at 2.01 eV. (b) Extracted energy shifts of the A-exciton, the lower and the upper hybrid interlayer exction as a function of the respective densities, showing enhanced nonlinearities in the presence of the external electric field. (c) Comparison of blueshifts of hybrid interlayer exction for the 0V and 1V cases. The hIE state under finite electric field shows a significantly stronger density dependence than in the degenerate case at 0V.
• Figure 4: Calculated exciton energy renormalization induced by interaction with a reservoir of hIE for the A exciton (a) and for the lower hIE (b). The effective exciton temperature is $70$ K. Multiple voltages are shown to visualize the trend with increasing external electric field.
• Figure 5: Individual contributions to the energy renormalization of the lower hIE corresponding to the total renormalizations shown in Fig. \ref{['fig:theory']}(b). The dipole-dipole contribution as direct part of the exciton Hartree self-energy is shown in (a) for different external electric fields. The Montroll-Ward contribution due to excitonic screening is shown in (b). The combined effect of fermionic exchange, contained as exchange terms in the Hartree and Fock self-energies, bosonic exchange contained as direct term in the Fock self-energy, as well as Pauli blocking is given in (c). As in Fig. \ref{['fig:theory']}, the density is distributed among the different hIE according to an effective temperature of $70$ K.